vectis lady - you should log the time as PU/T - I assume the other pilot was actually a FI? It would indeed be a good idea to have him/her sign your logbook column for rating revalidation purposes.
There is actually very little point in worrying about P1S time - since all licensing requirements specify PU/T time, PIC time and total time. PICU/S for successful check rides - fine. But although there is logic in recording flight by a pilot with a valid rating flying under the supervision of a FI ot CRI(SPA) as P1 under supervision, it won't actually matter in the big picture.... But it is NOT PIC time unless the other person is flying as a passenger.
A Club I know of was conducting 'check rides' on newly arrived pilots as a requirement they'd dreamt up - even though the pilot might have been current on the aircraft variant. The pilot being checked paid for the flight, logged it as PICU/S and the 'checking' pilot logged the time as P1C. But the 'checking pilot' held neither a FI rating nor CRI(SPA) rating....
